{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lab 2 slides

# Lab 2 slides - Review 1 N(1 2/SXX 1 0 N(0 2 n SSreg/1...

This preview shows pages 1–7. Sign up to view the full content.

Review ˆ β 1 N ( β 1 , σ 2 /S XX ) ˆ β 0 N ( β 0 , σ 2 ( 1 n + ¯ x 2 S XX )) SS reg / 1 SS res / ( n - 2) ∼ F 1 ,n - 2 Construction of t - intervals for β 1 and β 0 individually Lecture 4 – p. 1/2

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Assessing Fit Will later examine many different criteria for model fit and model selection Most basic: how much of the variability of the response is due to the variability in the covariate (or explanatory or independent) variable Natural measure is the coefficient of determination (or R 2 ): R 2 = SS Reg SS Tot Could also write as: R 2 = 1 - SS Res SS Tot Lecture 4 – p. 2/2
Properties of R 2 Obviously between 0 and 1 Larger values indicate more explanatory power of the covariate Does not reflect appropriateness of the model assumptions or the causal relationship of the covariate to the response (except in rare cases) Interpretation of R 2 varies depending on the scientific environment and application and goals of the analysis R 2 can be made artificially high by increasing S XX (and therefore the contribution of SS Reg to SS Tot ) with a few carefully chosen points (remember that E ( SS Reg ) = σ 2 + β 2 1 S XX Lecture 4 – p. 3/2

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Accounting data > summary(account.mod) Call: lm(formula = MARKETRATE ˜ ACCOUNTINGRATE) Residuals: Min 1Q Median 3Q Max -6.396 -3.307 -1.294 2.837 13.487 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.8480 1.9765 0.429 0.67 ACCOUNTINGRATE 0.6103 0.1432 4.263 8.5e-05 *** --- Signif. codes: 0 ’ *** ’ 0.001 ’ ** ’ 0.01 ’ * ’ 0.05 ’.’ 0.1 ’ ’ 1 Residual standard error: 5.086 on 52 degrees of freedom Multiple R-Squared: 0.259, Adjusted R-squared: 0.2448 F-statistic: 18.18 on 1 and 52 DF, p-value: 8.505e-05 Lecture 4 – p. 4/2
Two kinds of prediction Extremely important to distinguish between two types of predicted values Predicted mean of the response for a particular x 0 value Predicted value of the response for a particular x 0 value Although the numeric predictions are the same, the nature of uncertainty is very different Lecture 4 – p. 5/2

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Predicted mean response For any ˆ β 0 and ˆ β 1 and x 0 value we have ˆ y ( x 0 ) = ˆ β 0 + ˆ β 1 x 0 E y ( x 0 )) is obviously β 0 + β 1 x 0 by linearity of expectation Can show V ar y ( x 0 )) = V ar y + b 1 ( x 0 - ¯ x )) = σ 2 ( 1 n + ( x 0 - ¯ x ) 2 S XX ) NOTE: This gives point-wise confidence bands for the mean of the response, not a confidence band for the actual respone values!
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern